Stable and Efficient Computation of Generalized Polar Decompositions

TL;DR

This paper introduces a stable and efficient method for computing generalized polar decompositions using the DWH iteration, hyperbolic QR decomposition, and advanced Cholesky-based techniques, improving numerical stability especially for ill-conditioned matrices.

Contribution

It extends the DWH iteration to generalized polar decompositions with signature matrices and develops a stable CholeskyQR2-based approach with permuted graph bases for high accuracy.

Findings

Residuals of order 10^{-14} achieved for badly conditioned matrices

Stable implementation avoids matrix inversion using QR decompositions

Enhanced numerical stability with permuted graph bases

Abstract

We present methods for computing the generalized polar decomposition of a matrix based on the dynamically weighted Halley (DWH) iteration. This method is well established for computing the standard polar decomposition. A stable implementation is available, where matrix inversion is avoided and QR decompositions are used instead. We establish a natural generalization of this approach for computing generalized polar decompositions with respect to signature matrices. Again the inverse can be avoided by using a generalized QR decomposition called hyperbolic QR decomposition. However, this decomposition does not show the same favorable stability properties as its orthogonal counterpart. We overcome the numerical difficulties by generalizing the CholeskyQR2 method. This method computes the standard QR decomposition in a stable way via two successive Cholesky factorizations. An even better numerical stability is achieved by employing permuted graph bases, yielding residuals of order $10^{-14}$ even for badly conditioned matrices, where other methods fail.